View source for Abel transformation

''summation by parts''

A transformation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010190/a0101901.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010190/a0101902.png" /> are given, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010190/a0101903.png" /> is arbitrarily selected, and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010190/a0101904.png" /></td> </tr></table>

The Abel transformation is the discrete analogue of the formula for [[Integration by parts|integration by parts]].

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010190/a0101905.png" /> and if the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010190/a0101906.png" /> is bounded, then the Abel transformation can be applied to the series

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010190/a0101907.png" /></td> </tr></table>

The Abel transformation is used to prove several criteria of convergence of series of numbers and functions (cf. [[Abel criterion|Abel criterion]]). The Abel transformation of a series often yields a series with an identical sum, but with a better convergence. It is also regularly used to obtain certain estimates (cf. [[Abel inequality|Abel inequality]]), in particular, for investigations on the rate of convergence of a series. It was introduced by N.H. Abel [[#References|[1]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.H. Abel,   "Untersuchungen über die Reihe <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010190/a0101908.png" />"  ''J. Reine Angew. Math.'' , '''1'''  (1826)  pp. 311–339</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.I. Markushevich,   "Theory of functions of a complex variable" , '''1''' , Chelsea  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E.T. Whittaker,   G.N. Watson,   "A course of modern analysis" , '''1–2''' , Cambridge Univ. Press  (1952)</TD></TR></table>